That’s not quite how the London government saw it. You may not be familiar with King George’s August 1775 “Proclamation for Suppressing Rebellion and Sedition”. Other than the archaic language it sounds very much like Lincoln’s April 15, 1861 proclamation for 75,000 troops to suppress the rebellion.
This came after the 1774 ‘Petition to the King’ and the July 4, 1775 ‘Olive Branch Petition’ in which the Continental Congress stresses their loyalty as British subjects.
While perhaps interesting those comments ignore the critical difference in the situations.
We never consented to be subjects of the King. That is why we constituted a government to which we were to be citizens and to which we consented.
Similarity between those statements is not even interesting since the King could well have believed his words. And the president is an executive like the King.